Questions tagged [forcing]
Forcing is a set theoretic method used mainly for proving independence results. For questions about forcing function please use (differential-equations).
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"Correct" definition of $\kappa$-closed poset
Compare the following two definitions:
(1) a poset is $\kappa$-closed if any decreasing sequence of length $<\kappa$ has a lower bound;
(2) a poset is $\kappa$-closed if any decreasing sequence of ...
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How to force the unboundedness number to be uncountable?
Using Cohen forcing to add Cohen reals $c_\alpha$, $\alpha < \omega_1$, one can show that these form an unbounded family of said cardinality in the generic extension. As $\mathfrak{b} = \min \{ |\...
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Reading Intuitionistic Logic: Model Theory and Forcing in order to learn forcing [closed]
I'm trying to learn the intuitionistic approach to forcing by reading Intuitionistic Logic: Model Theory and Forcing by Melvin Fitting. I don't particularly care about the various theorems concerning ...
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$\mathbb{P}$-names for powersets in forcing extension
So, I am trying to understand how to solve the following:
Take the usual setup for forcing: a poset $\mathbb{P}$ lying in a countable transitive model $\mathcal{M}$. Give an explicit description of $\...
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If ($\mathbb{P} \ \text{is c.c.c.})^M$ then $\mathbb{P}$ preserves cardinals.
In "Old Kunen", we have that a partial order $\mathbb{P}$ preserves cardinals $$\iff \forall \beta \in o(M)[\beta > \omega \land (\beta \ \text{is a cardinal})^M \to (\beta \ \text{is a ...
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Choice of bound in the Laver property
$\newcommand{\P}{\mathbb{P}}$
Recall the following formulation of the Laver property (Page 459, Combinatorial Set Theory: With a Gentle Introduction to Forcing (Second Edition) by Halbeisen):
Laver ...
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Kunen recursive definition of $\Vdash^*$ in the $\exists$-case.
Stupid question: In Kunens "Set Theory An Introduction to Independence Proofs", VII, $\S3$, definition 3.3,(e), we are given the following: $$p \Vdash^* \exists x\ \phi(x,\tau_1,\ldots,\...
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Exercise IV 2.8 in "New Kunen" (Set Theory).
So, I am learning about the most basic setup to get forcing on the ground, and I feel like I want to understand the recursively defined notion \begin{align*} \text{val}(\tau,G) &:= \{\text{val}(\...
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Is there a structure or abstraction that genuinely generalizes both field extensions and (Cohen) forcing?
I'm just starting with the topic so I don't know much. But I'm always told that the similarity with field extensions is just an analogy and we shouldn't take it so seriously.
However, I just must ask ...
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Consistency Strength of OCA?
I was wondering whether we know the consistency strength of the Open Colouring Axiom (or Todorčević's Axiom). It is a consequence of PFA, so clearly it is below a supercompact, but I cannot find ...
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What exactly are characteristic functions of sets when constructing Boolean models?
Reading J. L. Bell's Set Theory --- Boolean-Valued Models and Independence Proofs. This is how the introduction of Boolean models is motivated:
Suppose that for each set $x ∈ V$ we are given a ...
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In what sense is forcing "impossible" in $L$?
I just saw an interesting video from Hugh Woodin about Ultimate $L$. In it, he says one of the reasons $L$ is so interesting is because it not only settles many natural set theory questions, but is ...
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Intuition for working with p-names.
So I have been trying to learn about forcing in set theory from kunen. There are these things called "p-names" which have a complicated definition in forcing. if $M$ is a countable ...
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What is meant by this association of the universe $V$ and the set of all formulas $\mathcal{L}^{(B)}$?
Reading J. L. Bell's Set Theory --- Boolean-Valued Models and Independence Proofs. The author introduces the class $V^{(B)}$, defined by
$$x \in V^{(B)} \iff \mathrm{Fun}(x) \land \mathrm{ran}(x) \...
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Introduction to Proper forcing
I’m trying to understand how proper forcing was introduced and the feeling of the model theoretic equivalence
Could you please recommend a great introductory text? I have read the Jech’s Multiple ...
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Can we sheaf-theoretically force a violation of the continuum hypothesis in a (nice) topos which is *bicomplete*?
$\newcommand{\p}{\mathcal{P}}$I recently dug through the exercises and details in Mac Lane and Moerdijk's book "Sheaves in Geometry and Logic" which concern themselves with (a baby version ...
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Cohen proof that : Forcing has to be done using ordinals within countable models
In Cohen "Set Theory and the Continuum Hypothesis" on page 110 says :
Our last theorem shows we must consider countable models. And then goes on to say :
M is a fixed countable model of ZF ...
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Which forcing technique implies "every set is countable from some perspective"? Which notion of "the same set" is used between models?
https://plato.stanford.edu/entries/paradox-skolem/ contains this claim:
Further, the multiverse conception leads naturally to the kinds of conclusions traditional Skolemites tended to favor. Let $a$ ...
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Infinite Set of Naturals in Forcing Extension
Just a random doubt that came up while talking to a friend. Assume we have some forcing $\mathbb P$ that behaves pretty nicely, so for example we can assume it doesn't collapse $\omega_1$, it doesn't ...
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Why is $[[x \in y]]$ not $y(x)$ in Boolean-valued models?
I'm reading about Boolean-valued models (in order to understand the first Cohen model), and it seems as though the truth value of the formula "$x \in y$", which is written $[[x \in y]]$, is ...
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What is the cardinality of $\operatorname{Add}(\lambda, \kappa)$ for $\kappa < \lambda$?
I am reading the proof of the theorem 6.17 (in the chapter VII page 215) of the Kunen's Book (the old edition), that says:
Theorem 6.17: In $M$, assume that $\lambda < \kappa$, $\lambda$ is ...
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Why is the union of working parts in the Hechler forcing a well-defined function?
Source: Set Theory, Third Edition by Thomas Jech.
Definition (Hechler Forcing): Let $\mathcal{G}$ be a given family of functions $h: \omega\to\omega$. A forcing condition is a pair $p=(s, E)$, where $...
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Kanamori, The Higher Infinite: Levy collapse, Propositions 10.20 and 10.21
I have a question regarding Kanamori's book The Higher Infinite, propositions 10.20 and 10.21.
Proposition 10.20 states:
Let $\mathbf{P} = (P, \leq)$ be a separative forcing notion with $\lvert P \...
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Does a set of ground models and forcing extensions of a c.t.m. can make a tree (set theory)?
I think the set of all such models become a partially ordered set. For instance, I define $M_1 \leq M_2$ as $M_2$ is a forcing extension of $M_1$ or reversibly, $M_1$ is a ground model of $M_2$. Then, ...
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If GCH is wrong in ZFC...
I am currently working on a presentation about the Symmetric Group $Sym(X):= \{ \sigma : X \to X \mid \sigma \text{ is bijective} \}$ of a set $X \neq \emptyset$.
I was able to show that for any ...
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Real degrees after forcing a random real
We know a lot about the properties of the real degrees if we assume the existence of an $L$-generic Cohen real (e.g. see Abraham and Shore Degrees of Constructibility of Cohen Reals). Among other ...
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Mapping Reflection Principle
I'm reading this paper written by Moore and he mentions the notion of a continuous $\in-$chain.
Can anyone tell me what he means by that? I can't find the definition
The link for the original paper ...
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Definition of a Partial Order (In: Kanamori, The Higher Infinite)
I have a question regarding a definition in Kanamori's book on page 136. (it's the beginning of the subsection about Random Reals)
Let $\mathcal{B}^{\star} = \{X \in \mathcal{B} \ \lvert \ X \ \text{...
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Infinite wacky race
Dick Dastardly is taking part in an infinite wacky race. What is infinite about it, you ask? Well, just everything! There are infinitely many racers, every one of which can run infinitely fast and the ...
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Is my intuition here regarding generic sets used in forcing correct?
In Jech's Chapter 14, p. 218 (2nd edition), the proof for the generic model theorem (i.e. Theorem 14.5) begins with: "Let $(P, <) $ be a notion of forcing in the ground model $M$, and let $G \...
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Forcing as a Quotient
I'm reading Jech and following his Boolean algebra models approach to it. I'm wondering if I've got the right idea here.
Let $M \models \mathrm{ZFC}$ and $B \in \mathbf{CompBoolAlg}$. We construct $M^...
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What are generic ideals and dense sets, intuitively?
Someone commented under one of my previous posts that, intuitively, a generic set isn't supposed to have any "conspicuous properties". I wonder what the precise meaning of that comment was ...
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Theorem of Shelah about the existence of an inaccessible cardinal
There is a theorem of Shelah, stated in the following way:
If all $\Sigma_3^1$ sets of reals are measurable, then $\aleph_1$ is an inaccessible cardinal in $L$.
In some textbooks (for example ...
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Forcing with the generic multiverse
Fix a countable transitive model $M$ and consider the collection $\mathbb{P}$ of all forcing extensions of $M$ (i.e., the generic multiverse of $M$), ordered by reverse containment. What happens if we ...
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Symmetric submodels of $M[G]$ are exactly the classes $(HOD(M[x]))^{M[G]}$
I'm trying to understand Grigorieff's proof (Theorem 3, page 478, Intermediate Submodels and Generic Extensions in Set Theory) that symmetric submodels of $M[G]$ are exactly $(HOD(M[x]))^{M[G]}$ for $...
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Questions about Namba forcing
I am looking at Jech's proof of properties of Namba forcing. The conditions consist of perfect subtrees of $\omega_2^{<\omega}$, where perfect means every node has $\omega_2$ many (not necessarily ...
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Where does the term 'dense' used in forcing/Martin's axiom come from?
There are some common meanings to 'dense' in Mathematics.
In Topology, a subset $S\subseteq X$ of a topological space $(X, \tau)$ is dense if the intersection of every non-empty open set with $S$ is ...
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BPI in Cohen Model
I'm reading Miroslav Repicky's paper "A proof of the independence of the Axiom of Choice from the Boolean Prime Ideal Theorem" and I'm confused by the claim that an ideal $I$ on a Boolean ...
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κ-chain-condition in Levy collapse
I'm studying the Levy collapse:
For S ⊆ On and λ regular, Col(λ, S) is defined in the following way:
Col(λ, S) = {p | p is a function ∧ |p| < λ ∧ dom(p) ⊆ S × λ
∧ ∀ <α, ξ > ∈ dom(p)(p(α, ξ ) =...
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Explicit construction of the transitive closure in the Boolean valued model
Suppose $t\in M^B$ is an element of the Boolean valued model. Since $ZF\vdash\exists$x x=TC({$Val_G(t)$}), there is $s\in M^B$ such that $1\Vdash Val_G(s)$=TC({$Val_G(t)$}). Is it possible to ...
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Confused by Definition 14.25 in Jech regarding generic ultrafilters
In Definition 14.25 of Jech's set theory, the canonical name $\dot{G}$ for a generic ultrafilter is the Boolean function defined by:
$$
\text{dom}(\dot{G})=\{\check{u} : u \in B \}
$$
such that:
$$
\...
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Motivation behind Generic filters in forcing.
So I was reading about forcing in wikipedia to try to get an intuitive idea about forcing in set theory. There is this paragraph in it:
A subtle point of forcing is that, if $X$ is taken to be an ...
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How to understand canonical ultrafilter in forcing chapter of Jech
Am new to forcing (obviously...), and I got to read Definition 14.25 in Jech's Set Theory which defines the canonical ultrafilter $\dot{G}$ on the Boolean algebra $B$ which is:
$\dot{G}(\check{u})=u$ ...
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A step in Kunen's proof of ccc preserving cofinalities
I have a question about one of the steps in the proof of "$ccc$ preserves cofinalities" in Kunen. The whole proof is below. My question concerns the claim that $\sup Y=\beta$. I am not sure ...
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How exactly does $G$ not being generic make $\mathbf{M}[G]$ not a model?
I've been trying to solve this exercise from Nik Weaver's Forcing for Mathematicians.
Let $P$ be the set of all finite partial functions from $\mathbb{N}^2$ into $\{0,1\}$, let $α$ be a countable ...
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Name for this Set
In Set Theory p. 209 - Jech (2003), the Jech performs the following construction:
Let $ B \in \mathbf{BoolAlg} $. For $ \alpha \in \mathbf{Ord} $ and $ \lambda \in \mathbf{Lim} $, let
$ V_0 = \...
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Permutation model in which infinite sets are Dedekind-infinite
I've been looking for a permutation model in which $IDI$ (every infinite set is Dedekind-infinite) and $\neg AC$ holds. I have found several (for example in Countable Sums and Products of Metrizable ...
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Is this exercise on forcing trivial?
I'm reading Nik Weaver's Forcing for Mathematicians. This is one of the exercises.
Let $P$ be a forcing notion, let $G$ be a generic ideal of $P$, and suppose $G \in \mathbf{M}$. (As we noted at the ...
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Finite/Countable support iteration of countable cofinality and mad families
Using a finite support iteration $\langle P_\alpha, \dot{Q}_\alpha : \alpha < \kappa\rangle$ it has been shown that if $\kappa$ has uncountable cofinality, then using Mathias forcing one obtains a ...
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Is every ccc Baire forcing a Suslin tree?
Exercise IV.6.11 of Kunen says there exists an atomless ccc Baire poset iff there is a Suslin tree; I think here Baire can be taken to mean either countable intersection of dense open sets is dense or ...
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